Film thickness distribution in gravity-driven pancake-shaped droplets rising in a Hele-Shaw cell

Abstract

We study here experimentally, numerically and using a lubrication approach; the shape, velocity and lubrication film thickness distribution of a droplet rising in a vertical Hele-Shaw cell. The droplet is surrounded by a stationary immiscible fluid and moves purely due to buoyancy. A low density difference between the two mediums helps to operate in a regime with capillary number Ca lying between 0.03 − 0.35, where Ca = µ o U d /γ is built with the surrounding oil viscosity µ o , the droplet velocity U d and surface tension γ. The experimental data shows that in this regime the droplet velocity is not influenced by the thickness of the thin lubricating film and the dynamic meniscus. For iso-viscous cases, experimental and three-dimensional numerical results of the film thickness distribution agree well with each other. The mean film thickness is well captured by the Aussillous & Quéré (2000) model with fitting parameters. The droplet also exhibits the "catamaran" shape that has been identified experimentally for a pressure-driven counterpart (Huerre et al. 2015). This pattern has been rationalized using a two-dimensional lubrication equation. In particular, we show that this peculiar film thickness distribution is intrinsically related to the anisotropy of the fluxes induced by the droplet's motion.

Motion of a confined bubble in a shear-thinning liquid

“…The seminal work of Bretherton (1961) demonstrated that h/R = 1.34 Ca 2/3 in the limit of small Ca, as supported by several measurements (Schwartz, Princen & Kiss 1986;Aussillous & Quéré 2000). Thereafter, many researchers investigated the effect of finite capillary number and inertia (Cox 1962;Reinelt & Saffman 1985;Aussillous & Quéré 2000;Heil 2001;de Ryck 2002;Khodaparast et al 2015;Magnini et al 2017), surfactants and variable surface tension, (Ratulowski & Chang 1990;Park 1992;Stebe & Barthés-Biesel 1995;Olgac & Muradoglu 2013;Yu, Khodaparast & Stone 2017), unsteady flow, (Yu et al 2018), buoyancy, (Leung et al 2012;Atasi et al 2017;Lamstaes & Eggers 2017) and bubble viscosity, (Chen 1986;Hodges, Jensen & Rallinson 2004;Balestra, Zhu & Gallaire 2018;Shukla et al 2019).…”

“…2017; Lamstaes & Eggers 2017) and bubble viscosity, (Chen 1986; Hodges, Jensen & Rallinson 2004; Balestra, Zhu & Gallaire 2018; Shukla et al. 2019).…”

Liquid velocity fluctuations and energy spectra in three-dimensional buoyancy-driven bubbly flows

“…For small Atwood numbers, we employ the Boussinesq approximation, whereby ρ on the left-hand side of (2.1a) is replaced by the average density ρ a . Note that low Atwood numbers can be experimentally realized in near-critical binary fluids as well as mixtures of oil (Perlekar 2019;Shukla et al 2019).…”

Taylor drop in a closed vertical pipe

“…When viscous forces dominate over inertial and surface tension effects, the ascent speed of a gaseous Taylor bubble is proportional to ΔρgD 2 /μ d , where μ d is the viscosity of the liquid (Brown 1965;Wallis 1969). In contrast, when surface tension dominates over buoyancy, the ascent speed and film thickness are controlled by capillary forces (Bretherton 1961;Reinelt 1987;Batchelor 2000;Llewellin et al 2012;Shukla et al 2019).…”